Download - Advance logic
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Overview
What is functional decomposition Why functional decomposition Decomposition using decomposition chart Decomposition using BDD
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Functional decomposition
Functional decomposition refers to the process by which a complex problem or system is broken down into parts that are easier to conceive, understand and maintain.
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Functional Decomposition
functional decomposition generally refers to a process of identifying a set of functions such thatƒ(x,y) = h(g1(x), g2(x), ……, gn(x), y)
ƒg1
hgn
X
Y
X
X
Y
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Why functional decomposition
Requires less number of logic gates than the original networks.
Reduces the complexity of the circuit Parallel execution is possible for the different
module of the function
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Procedure of Decomposing a Function
Construct decomposition charts for all possible bipartition of the inputs.
Determine the column multiplicity of each chart. Calculate the gain, if gain > 1 then the function is
decomposable. We will choose that particular bipartition, for
which gain is maximum. Construct the logic circuit applying
decomposition of the given function
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Functional Decomposition using Decomposition Chart
X1
X2
X3
X4
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
ƒ 1 0 0 1 1 0 0 1 1 1 0 0 1 0 0 1
00 01 10 11
00 1 1 1 1
01 0 0 1 0
10 0 0 0 0
11 1 1 0 1
X1X2X3X4
Figure: Decomposition Chart
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Decomposition Chart
00 01 10 11
00 1 1 1 1
01 0 0 1 0
10 0 0 0 0
11 1 1 0 1
The number of distinct columns in the decomposition chart is called Column Multiplicity
Here Column multiplicity is 2
X1X2
X3X4
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Decomposition Chart
1 1
0 1
0 0
1 0
00
01
10
11
000111
10 Bound Set (X1,X2)
Free Set (X3,X4)
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Condition for Decomposition
A function f is decomposable if we get at least one non-trivial decomposition for the function.
The decomposition is said to be non-trivial if Gain > 1.
Gain = min(2n1,22*n2) / µ
Where, n1 = bound Set n2 = free Setµ = column multiplicity
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Example
x4x5
000
001
010
011
100
101
110
100
00 0 1 0 0 0 1 1 0
01 1 1 1 1 1 1 1 1
10 1 0 1 1 1 0 0 1
11 0 1 0 0 0 1 1 0
x3x4x500011011
000 0 1 1 0
001 0 1 1 1
010 1 0 0 0
011 1 1 1 1
100 0 0 0 1
101 0 1 1 0
110 1 0 0 1
111 1 0 1 1
Gain = MIN(22,22*3) / 3 = 1.33
Gain = MIN(23, 22*2) / 2 = 4
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Decomposition using BDD
•BDD is a directed acyclic graph which represents a Boolean function•Each node represents a sub-function and two incoming edges of this node represent cofactors of this function
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0 1
0 1 0 1
0 10 1
0 1 0 1
0 1
0 1
0 1
X4
X3
X2
X1
x4x3x2 + (x4 + x3 + x2) x1
x2x1
(x3 + x2) x1 x3x2 + x1
x2 + x1 x1
x10 1
Decomposition using BDD
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0 1
0 1 0 1
0 10 1
0 1 0 1
0 1
0 1
0 1
X4
X3
X2
X1
x4x3x2 + (x4 + x3 + x2) x1
x2x1
(x3 + x2) x1 x3x2 + x1
x2 + x1 x1
x10 1
Decomposition using BDD
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0 1
0 1 0 1
0 10 1
0 1
0 1
X4
X3
X2
X1
x4x3x2 + (x4 + x3 + x2) x1
x2x1
(x3 + x2) x1 x3x2 + x1
x2 + x1
x1
ROBDD
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Thank You